Compact and Tunable Transmitter and Receiver for Magnetic

0

Compact and Tunable Transmitter and Receiverfor Magnetic Resonance Power Transmission to

Mobile Objects

Takashi Komaru, Masayoshi Koizumi, Kimiya Komurasaki,Takayuki Shibata and Kazuhiko Kano

DENSO CORPORATION and the University of TokyoJapan

1. Introduction

As electronic devices are becoming more mobile and ubiquitous, power cables are turningto the bottlenecks in the full-fledged utilization of electronics. While battery capacities arereaching their limits, wireless power transmission with magnetic resonance is expected toprovide a breakthrough for this situation by enabling power feeding available anywhere andanytime.This chapter studies the feasibility of magnetic resonance power transmission to mobileobjects mainly focusing on the resonator quality factor and impedance matching controlsystems. Transmission efficiency reaches a reasonably high level when the transmitting andreceiving resonators satisfy two conditions. The first is to have high quality factors. Thesecond is to tune and match the impedance to the transmission distances. The second sectionexplains the theoretical grounds for these conditions. The third section describes a developedwireless power transmission system prototype which was made compact and tunable to beapplied to mobile objects. The later sections evaluate the quality factor and the impedancematching of the prototype.

2. Theoretical analysis of a magnetic resonance system

This section states the theoretical basics of wireless power transmission with magneticresonance. The theory is developed by using the logic of electrical engineering without thecoupled-mode theory. First, the base concept of a wireless power transmission system andits equivalent circuit model are introduced. Then the transmission efficiency is derived as aformula with the physical properties including the impedances and the mutual inductance.This formula is simplified by replacing the physical properties with the non-dimensionalparameters including impedance ratio, quality factor and coupling coefficient. Analysis ofthis simplified formula derives the essential principle for efficient mid-range wireless powertransmission.

2.1 Basic model

The model of a wireless power transmission system with magnetic resonance is expressed asper Fig. 1 (a) and (b). Two resonators of a series LCR (inductor, capacitor and resister) are

7

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2 Will-be-set-by-IN-TECH

inductively coupled to each other. Note that this model also applies to the basic system ofwireless power transmission with electromagnetic induction. The currents flowing throughthe source and load are derived from Kirchhoff’s second law as per (1).

[

Vsrc

0

]

=

[

ZS jωMSD

jωMSD ZD + Z0ld

] [

Isrc

Ild

]

⇒[

Isrc

Ild

]

=Vsrc

ZS(ZD + Z0ld) + (ωMSD)2

[

ZD + Z0ld

−jωMSD

]

(1)

Transmittingresonator(S)

Receivingresonator(D)Power

source(src)

Load(ld)

(a) Image of the basic system

RS

MSD

CS RD CD LDLS

R ld=Z 0ld

Is rc Ild

Vs rc

(b) Equivalent circuit model

Powersource(src)

Load(ld)Vi

VrVt

Z0src Z0ld

Transmitter &Receiver

(c) Two-port network unit model

Fig. 1. Three types of concept figures of the basic wireless power transmission system model

Note that ω represents the angular frequency. Voltage Vsrc and currents Isrc, Ild are inphasor form and are complex properties. Impedance ZS is defined as the impedance of thetransmitting resonator RS + j(ωLS1/ωCS). Impedance ZD is also defined in the same wayas ZS. Z0src represents the characteristic impedance of the transmission line in the powersource device and the cable between the source and transmitting resonator. Impedance Z0ld

represents the characteristic impedance of the transmission line in the cable between the loadand receiving resonator. These characteristic impedances are assumed to be a real number.

134 Wireless Power Transfer – Principles and Engineering Explorations

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Compact and Tunable Transmitter and Receiver for Magnetic Resonance Power Transmission to Mobile Objects 3

The load resistance Rld is matched to Z0ld so that the load absorbs all transmitted power andreflects no power to the receiving resonator.Since the source frequency is often set to a high value of about 10 MHz, the concepts ofincident wave, reflected wave and transmitted wave should be introduced (Pozer, 1998). AsFig. 1 (c) shows, the coupled resonators inserted between the source and the load can beconsidered as a two-port network unit. This unit receives incident waves from the source.At the same time, it produces reflected waves to the source and transmitted waves to theload. Voltage Vi, Vr and Vt represent the amplitude of each wave respectively. And they areexpressed by the properties of the circuit model as per (2)

Vi =Vsrc + Z0src Isrc

2, Vr =

Vsrc − Z0srcIsrc

2, Vt = Z0ld Ild (2)

The power of each wave is as per (3).

Pi =|Vi|2

2Z0src, Pr =

|Vr|22Z0src

, Pt =|Vt|22Z0ld

(3)

Then the transmission efficiency is defined as the ratio of transmitted wave power andincident wave power.

η =Pt

Pi=

4Z0srcZ0ld(ωMSD)2

|(ZS + Z0src)(ZD + Z0ld) + (ωMSD)2|2 (4)

The reflection efficiency is also defined as the ratio of reflected wave power and incident wavepower.

ηr =Pr

Pi=

∣

∣

∣

∣

(ZS − Z0src)(ZD + Z0ld) + (ωMSD)2

(ZS + Z0src)(ZD + Z0ld) + (ωMSD)2

∣

∣

∣

∣

2

(5)

Equation (4) is the fundamental definition of efficiency. However, in some cases such as lowfrequency operations, the power source can reuse the reflected wave. Then the transmissionefficiency becomes η′ = η/(1 − ηr), which is equal to the efficiency derived from a simple ACcircuit calculation as per (6). Note that θ represents the phase difference between Vsrc and Isrc.

η′ =η

1 − ηr=

Z0ld|Ild|2|VsrcIsrc| cos θ

=Z0ld|Ild|2

RS|Isrc|2 + (RD + Z0ld)|Ild|2(6)

2.2 Analysis of transmission efficiency

The following non-dimensional parameters are useful when considering the transmissionefficiency analytically.

• Coupling coefficient k = MSD/(LSLD)1/2

• Quality factor QS = ω0LS/RS and QD = ω0LD/RD

• Impedance ratio rS = Z0src/RS and rD = Z0ld/RD

135Compact and Tunable Transmitter and Receiver for Magnetic Resonance Power Transmission to Mobile Objects

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Note that the resonant frequencies are assumed to be equal to each other ((LSCS)−1/2 =

(LDCD)−1/2 = ω0) for simplicity in this discussion. The frequency band is considered

narrow enough (|ω − ω0| ≪ ω0). Then (4) is expressed as (7) by using the approximationsω/ω0 − ω0/ω ≈ 2(ω − ω0)/ω0 and ω ≈ ω0.

η ≈4k2 rS

QS

rDQD

[

k2 − 4(

ω−ω0ω0

)2+ 1+rS

QS

1+rDQD

]2

+ 4(

ω−ω0ω0

)2 (1+rSQS

+ 1+rDQD

)2(7)

Efficiency η′ is also expressed by the non-dimensional parameters.

η′ =k2QSQDrD

(1 + rD)(1 + rD + k2QSQD) + 4[(ω − ω0)/ω0]2Q2D

(8)

In the ω − rS − rD domain, η has its maximum value (9) at (10).

ηmax =k2QSQD

(

1 +√

1 + k2QSQD

)2(9)

ω = ω0, rS = rD =√

1 + k2QSQD (10)

The physical meaning of (10) is the impedance matching condition. In the case rS = rD , ηin the ω − rS − rD domain becomes visible as the surface in Fig. 2, which clearly shows theexistence of a maximum point. It is not easy to prove directly from (7) that (9) is exactly themaximum. However, there is a simple and strict proof via analysis at (8).

η′ ≤ k2QSQDrD

(1 + rD)(1 + rD + k2QSQD)(11)

≤ k2QSQD

(1 +√

1 + k2QSQD)2≡ η′

max (12)

The equality in (11) and (12) is approved when ω = ω0 and rD = (1 + k2QSQD)1/2

respectively. These directly mean the maximum of η′ is η′max = ηmax at ω = ω0 and

rD = (1 + k2QSQD)1/2. Besides η ≤ η′ because η′ ≤ η/(1 − ηr) and 0 ≤ ηr ≤ 1. Thus

the maximum value of (7) is proved to be (9) as per (13).

η ≤ η′ ≤ η′max = ηmax ⇒ η ≤ ηmax (13)

Equation formula (9) depends only on one term k(QSQD)1/2 and draws a simple curve as

shown in Fig. 3.Here the principles for efficient power transmission are derived without referring to thecoupled-mode theory.


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0.991

1.010

10

20

0

20

40

60

80

100Maximum:'max

)=)0

rS = rD = (1+k2QSQD)1/2T

ran

smis

sio

ne

ffic

ien

cy,

'[%

]

Frequency ratio,)/)0

Impedance ratio, rS = rD

80 %60 %

40 %20 %

Fig. 2. Three dimensional plot and contour of the transmission efficiency in ω − r domainwith k = 0.01 and QS = QD = 1000.

0

20

40

60

80

100

0.1 1 10 100

Ma

xim

um

tra

nsm

issi

on

eff

icie

nc

y,

8m

ax

[%]

Figure of merit, k(QSQD)1/2

Fig. 3. Maximum transmission efficiency plotted along figure-of-merit

• k(QSQD)1/2 is defined as figure-of-merit (fom).

• Impedance ratio should be matched to (1 + fom2)1/2.

• Figure-of-merit in 1 or higher order enables efficient wireless power transmission withover 20% efficiency. (The concept of a strong coupling regime.)

They completely correspond to the characteristics of mid-range transmission argued in theoriginal theory of magnetic resonance (Karalis et al., 2008). In fact, k(QSQD)

1/2 can be derivedfrom the original expression of figure-of-merit κ/(ΓSΓD)

1/2 (Kurs et al., 2007). In the fieldof electrical engineering, coupling coefficient k and the quality factor Q are more intuitiveparameters than the reciprocally-time-dimensional coupling coefficient κ and decay constantΓ. Thus the redefinition of figure-of-merit helps the understanding of mid-range transmissionin comparison with conventional close-range electromagnetic inductance.In close-range electromagnetic inductance, k is over 0.8 in most cases and rarely less than 0.2.Therefore, it is relatively easy to achieve high transmission efficiency of over 80% (Ayano et al.,2003; Ehara et al., 2007). Meanwhile in the mid-range, the transmission distance is equal to orgreater than the size of resonators and k becomes a very low value of 102 or lower order. Here,the principle of the strong coupling regime concept leads to the point of importance: it is ahigh quality factor of 102 or greater order that enables efficient transmission even with verylow k in mid-range.In addition to high quality factor, impedance matching is needed for efficient transmission inmid-range. Mid-range wireless power transmission is expected to bring much more flexibility


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in receiver and transmitter positioning than close range wireless power transmission. Andthis means the coupling coefficient between transmitting and receiving resonators can takevarious values. Moreover in mobile applications, receivers and transmitters will be equippedon moving objects and have to exchange power across ever-changing transmission distanceand coupling coefficients. As stated before, the impedance ratio rS and rD must be matched to(1 + f om2)1/2 = (1 + k2QSQD)

1/2 in order to obtain the highest efficiency. Fig. 4 shows howtransmission efficiency changes with different coupling coefficients. When the receiver comesto the closer (a) or the farther (c) position than the original position (b), coupling coefficientbecomes larger or smaller and the maximum point in ω-r domain changes to the higher-r orthe lower-r point. Thus we must consider a system that controls the impedance ratio accordingto the variable transmission distance.There are three schemes considered for the impedance matching control: two-sided, one-sidedand no control. In the two-sided control scheme, both the transmitter and the receiver takethe optimum impedance ratios. In the one-sided control scheme, either the transmitter or thereceiver takes the optimum one and the other takes a fixed one. In the no control scheme, boththe transmitter and the receiver take fixed ones. The theoretical efficiencies with these controlschemes are expressed as follows:

η2 =k2QSQD

(1 +√

1 + k2QSQD)2(14)

η1 =k2QSQDrD

(1 + rD)(1 + rD + k2QSQD)(15)

η0 =4k2QSQDrSrD

[(1 + rS)(1 + rD) + k2QSQD]2(16)

Note that η2, η1 and η0 represent the efficiency with two-sided, one-sided and no controlscheme respectively. Equation 15 assumes that the transmitter has the optimized impedanceratio and the receiver has the fixed one. When the transmitter has the fixed one and thereceiver has the optimized one in reverse, rD and rS switch positions with each other. The nocontrol efficiency η0 is derived from η in (7) by substituting ω with ω0. The one-sided controlefficiency η1 is derived from the partial differential analysis of η0 as follows.

∂

∂rSlog ηnctl = 0 ⇒ k2QSQD + (1 − rS)(1 + rD) = 0 (17)

Efficiency η1 is equal to η′ in (8) with ω = ω0. This means the reuse of the reflected powerby the power source is equal to the optimization of the transmitter impedance ratio. Thetwo-sided control efficiency η2 is exactly the same as ηmax.The characteristics of the transmission efficiencies for the two-sided, one-sided and no controlare visualized by Fig. 5. Generally, two-sided control always takes the maximum efficiency,while it will take more hardware and software cost for the control system. No control willonly need a simple and low-cost system in exchange for efficiency at various transmissiondistances. One-sided control has the middle characteristics.


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0.991

1.010

10

20

0

20

40

60

80

100

59.3 %

Tra

nsm

issi

on

eff

icie

ncy

,8

[%]

Frequency ratio,?/?0Impedance ratio, rS = rD

80 %

60 %40 %20 %

Maximum: 90.5 %

21.2 %

(a) k = 0.02

0.991

1.010

10

20

0

20

40

60

80

100

Tra

nsm

issi

on

eff

icie

ncy

,5

[%]

Frequency ratio,=/=0Impedance ratio, rS = rD

80 %60 %

40 %20 %

54.4 %

54.4 %

Maximum: 81.9 %

(b) k = 0.01

0.991

1.010

10

20

0

20

40

60

80

100

Tra

nsm

issi

on

eff

icie

ncy

,5

[%]

Frequency ratio,=/=0Impedance ratio, rS = rD

60 % 40 % 20 %

46.7 %

18.3%

Maximum: 67.2 %

(c) k = 0.005

Fig. 4. Three dimensional plot and contour of transmission efficiencies in ω − r domain withvarious coupling coefficients k and quality factors QS = QD = 1000


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0

20

40

60

80

100

0.0001 0.001 0.01 0.1 1

Effic

iency [%

]

Coupling coefficient, k

Two-sided control, 2One-sided control, 1

No control, 0

Fig. 5. Transmission efficiency with the three types of impedance matching control schemes.The quality factors are QS = QD = 1000. The impedance ratios are fixed asrS = [1 + (10−2)2QSQD]

1/2 in the one-sided control and rS = rD = [1 + (10−2)2QSQD]1/2 in

the no control.

3. Specifications of the transmitter and the receiver

Efficient mid-range wireless power transmission with magnetic resonance needs a highquality factor resonator and impedance matching system. Mobile object applications needcompact and tunable transmitters and receivers. We developed an experimental transmitterand receiver system that satisfy those conditions as shown in Fig. 6. The system consistsof a resonator and a pickup loop. The resonator is a copper wire loop with a lumpedmica capacitor. The pick up loop is also a copper wire loop with a lumped mica capacitor.The combination of the resonator and the pickup loop function as an inductive transformer,which virtually transforms the characteristic impedance of the source and the load. Both theresonator and the wire loop have only a single turn and they are placed in parallel and veryclose to each other. This structure makes the transmitter and receiver axially compact. Notethat the resonant frequencies of the resonator and the pickup loop are designed to be equal toeach other. The following two sections state the performance analyses of the system.

4. Resonator quality factor

Resonators may take any shape and any materials. A popular type of resonator is a coil withmultiple turns. To make it compact, the turn pitch has to be as small as possible. But withsuch a dense coil structure, it is hard to have high quality factor because of the electricity lossdue to the insulating cover on the wire (Komaru et al., 2010). The other widely consideredtype is a coil with a single turn and a lumped capacitor. In this study, we call this type "loopwith capacitor" and analyze it as a primary model of the resonator.

4.1 Theoretical calculation

An exact calculation of quality factors is indispensable to design and evaluate the wirelesspower transmission system with magnetic resonance. As stated in the previous section, aresonator is equivalent to a series LCR element and quality factor Q is generally derived as


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Source

(src)Load

(ld)

Transmitting resonator

(S)

Receiving resonator

(D)

Receiving pickup coil

(B)

Transmitting pickup coil

(A)

(a) Overall view of the power transmission system

l

lA A'

a

A - A'

(b) The resonator with side length ℓ× ℓ = 198mm × 198 mm, wire radius a = 1.5 mm andcapacitor capacitance: C = 200 pF.

lA A'

a

A - A'l

y

(c) The pickup loop with side length ℓ × ℓ =92 mm × 92 mm, wire radius a = 1.0 mm,capacitor capacitance C = 470 pF and slidingposition y = 0 - 48 mm.

yS

zSA

S

A

yD

zDB

D

B

z

(d) Alignment of the resonators and pickup loopswith the pickup loop slide positions yS, yD = 0- 48 mm and the spaces between resonator andpickup loop ZSA = ZDB = 2.0 mm.

Fig. 6. Transmitter and receiver specification


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the ratio of resonant frequency f0 = ω0/2π = 1/2π(LC)1/2, self inductance L and resistanceR of a resonator.

Q =ω0L

R(18)

Of these properties, resistance is the most difficult to calculate regarding its realizable value.The predictions in most conventional research only take account of two types of resistance:radiation and ohmic resistance. This chapter also calculates capacitor resistance to predictthe quality factor precisely. Then the properties to be calculated are resonant frequency,self inductance, radiation resistance ohmic resistance and capacitor resistance. The first fourproperties are calculated by the theoretical equations derived from the electromagnetism. Theother capacitor resistance is estimated by using the measured specification data of a capacitorelement.The total length of the resonator wire is about 0.8 m and is far smaller than the wavelengthof about 30 m in the supposed frequency band. Thus the current distribution on the wire isapproximated to be uniform hereinafter.For a rectangle loop with a side length ℓx , ℓy and a wire radius a, the self inductance isexpressed as (19) (Grover, 1946).

L =μ0

π

[

ℓx ln

(

2ℓx

a

)

+ ℓx ln

(

2ℓy

a

)

+ 2 ·√

ℓ2x + ℓ2

y

−ℓx sinh−1(

ℓx

ℓy

)

− ℓy sinh−1(

ℓy

ℓx

)

− 1.75(ℓx + ℓy)

]

(19)

Substituting ℓx and ℓy with ℓ leads to

L =2μ0ℓ

π

[

ln

(

2ℓ

a

)

− 1.2172

]

(20)

The uniform current distribution means zero electrical charge density at every point on thewire. Therefore the resonator capacitance originates just from the lumped capacitor. Then theresonant frequency is

ω0 =1√LC

(21)

The resonator is approximated as a small current loop, which is theoretically equal to a smallmagnetic dipole as in Fig. 7, because of the loop size being far smaller than the wavelength.The radiation resistance is derived from consideration of this small magnetic dipole model. Asmall electrical dipole with current I and length dz radiates power expressed as

π|I|23

√

μ0

ǫ0

(

dz′

λ0

)2

(22)

Note that λ0 is the vacuum wavelength c0/ f0 = 2πc0/ω0. According to the symmetry ofMaxwell’s equations, ǫ0, μ0 and I are the dual of μ0, ǫ0 and Im respectively. Im represents


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dz'

Im

+Qm

-Qm

I

dS'

Fig. 7. Magnetic dipole and electrical current loop

the magnetic current (Johnk, 1975). Then the radiated power from a small magnetic dipole isexpressed as

Prad =π|Im|2

3

√

ǫ0

μ0

(

dz′

λ0

)2

(23)

From the definition of the magnetic dipole moment of both the magnetic dipole and theelectrical current loop

Qmdz′ = μ0 IdS′ (24)

where dS′ is the loop area corresponding to ℓ2 and Qm is the magnetic charge with the relation

Im =∂Qm

∂t= jωQm (25)

The radiated power is then transformed to

Prad =μ0ω4ℓ4

12πc30

|I|2 (26)

which should be equal to

Prad =1

2Rrad|I|2 (27)

Hence the radiation resistance is

Rrad =μ0ω4ℓ4

6πc30

(28)

The basic expression of the ohmic resistance for a wire with conductivity σ, sectional area Sand length dℓ is

Rohm =dℓ

σS(29)

When the frequency is high, current density decays exponentially along the depth from thesurface to the center of the wire. The skin depth δs is defined as the inverse of the decayconstant (Pozer, 1998) as

δs =

√

2

ωμσ(30)


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0

0.0005

0.001

0.0015

0.002

0.0025

0.003

10 11 12 13 14 15

Lo

ss

co

eff

icie

nt,

D

Frequency, f [MHz]

MeasuredApproximated line

Fig. 8. Characteristics of the capacitor loss coefficient

and wire sectional area S in (30) is substituted with S = 2πaδs

Rohm =dℓ

2πa

√

μω

2σ(31)

the total wire length of the resonator is 4ℓ and the ohmic resistance finally becomes

Rohm =ℓ

πa

√

2μω

σ(32)

A real capacitor is not a pure capacitive element but a complex component with variousimpedance elements. In the frequency band of wireless power transmission, the dominantcharacteristics are the original capacitance and the equivalent series resistance (ESR). Thisstudy calls ESR "capacitor resistance" and analyzes its value. The characteristics of the actualcapacitor were measured by the LCR meter. The meter gave the loss coefficient data as in Fig.8. Note that the approximated line is derived by using the least-squares method. The losscoefficient of the capacitor is defined as

D(ω) = − R(ω)

X(ω)= ωCR (33)

Note that X and R here represent the reactance and the resistance of the capacitor respectively.The capacitor resistance is then expressed as

Rcap =D(ω)

ωC(34)

4.2 Experimental measurement

The comparison with the calculated and the measured quality factors are shown in Fig. 9in the form of the loss coefficient. Note that the loss coefficient is the inverse of the qualityfactor. The calculated loss coefficient is classified into the three categories corresponding tothe three resistances introduced in the previous subsection. The radiation loss is too small tobe visible in this graph. The predicted resonant frequencies and the quality factors are wellagreed with the measured parameters, verifying the calculation method stated in this section.The important point is the quality factor of the loop with the capacitor type resonator was


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0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004

Measured(Transmitter)

Calculated

Loss coe cient, 1/Q

Measured(Receiver)

Ohmic losscoefficient

Capacitor losscoefficient

Q = 308

Q = 323

Q = 323

Fig. 9. Comparison between the calculated and the measured quality factors

higher than that of the dense coil type resonator at about Q = 200 as measured in our previousstudy. And the capacitor loss was as large as the ohmic loss. This means that the calculationof the capacitor loss was indispensable. Note that the measured resonant frequencies of thetransmitting and the receiving resonator were 13.43 and 13.46 MHz. They also agreed withthe calculated value of 13.54 MHz.

5. Impedance matching control system

Impedance matching means the transformation of the source and the load impedance. Theimpedance matching system has the same purpose as the antenna tuners. There are severalways to implement impedance matching: some use a set of variable capacitors and inductorsand some others use transistors for example. The main requirement of the impedancematching system for efficient transmission is low electrical loss. And for mobile applications,it is also required to be compact and have wide range of impedance transformation. Thischapter considers a sliding pickup loop system as a simple and compact transmitter andreceiver prototype.

5.1 Basics of impedance matching with pickup loop

Impedance transformation is explained as follows. The coupling of the resonator and thepickup loop is equivalent to the circuit model shown in Fig. 10. Though the indices here are forthe receiving resonator and the pickup loop, the same result is obtained for the transmittingresonator and the pickup loop. And the following mutual inductances are assumed to benegligible: MAB between the transmitting and the receiving pickup loop, MSB betweenthe transmitting resonator and the receiving pickup loop and MDA between the receivingresonator and the transmitting pickup loop. From Kirchhoff’s second law for the circuit, thecurrents flowing in the resonator and the pickup loop ID, Ild is expressed by the voltage in theresonator V which the current in the pickup loop inducted.

[

V0

]

=

[

0 jωMDB

jωMDB ZB + Z0ld

] [

ID

Ild

]

⇒[

ID

Ild

]

=V

(ωMDB)2

[

ZB + Z0ld

−jωMDB

]

(35)


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RD

MDB

CD RB CB LBLD

Z0ld

ID

Ild

V

RD CD LD

ID

Z'0ld

Fig. 10. Circuit model of the impedance transformation with pickup loop

Then the pickup loop and the load that are inductively coupled to the resonator equal to theimpedance Z0ld directly connected to the resonator.

Z′0ld =

V

ID=

(ωMDB)2

ZB + Z0ld(36)

Note that ZB represents the impedance of the pickup loop ZB = RB + i(ωLB1/ωCB). Let theresonant frequency of the pickup loop equal the resonant frequency of the resonator and thesource frequency so that the pickup loop reactance is zero ZB = RB. Now the impedance ratiobecomes

rD =Z′

0ld

RD=

V

RD ID=

(ωMDB)2

RD(RB + Z0ld)(37)

This is the function of the mutual inductance between the resonator and the pickup loop.This means it is possible to control the impedance ratio by changing the coupling condition,including the relative position, of the resonator and the pickup loop.To design a transmitter or a receiver with this pickup loop impedance matching system andacquire the desired impedance ratios, the following properties have to be calculated: theresistance RB, the self inductance LB, the capacitance CB of the pickup loop and the mutualinductance between the resonator and the pickup loop MDB. For the pickup loop with asquare loop wire and a lumped capacitor, the first three properties are calculated by the samemethod introduced in previous section. The last property is calculated from electromagnetictheory as explained in the following subsection.

5.2 Mutual inductance between square loops

For coupling of square loop wires, the theoretical formula of mutual inductance is derivedfrom the Neumann formula (38), which expresses the mutual inductance of coupled circuitss1 and s2 in free space, as illustrated in Fig. 11 (a).

μ0

4π

∫

s1

∫

s2

d�s1 · d�s2

|�s1 − �s2|(38)

Then the mutual inductance between two parallel wires of finite line, as shown in Fig. 11 (b),is derived by using the Neumann formula as


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M‖(ℓ1, ℓ2, ξ, ζ) = m‖(ℓ1, ℓ2, ξ, ζ) + m‖(ℓ1, ℓ2,−ξ, ζ)

−m‖(ℓ1,−ℓ2, ξ, ζ)− m‖(ℓ1,−ℓ2,−ξ, ζ) (39)

The abbreviation m‖ is defined as

m‖(ℓ1, ℓ2, ξ, ζ) ≡ μ0

8π

[

(ℓ1 + ℓ2 + 2ξ) sinh−1 ℓ1 + ℓ2 + 2ξ

2ζ−√

(ℓ1 + ℓ2 + 2ξ)2 + (2ζ)2

]

(40)

Now consider the couple of parallel square loop illustrated in Fig. 11 (c). The couple consistsof 8 lines: line a to d and A to D. According to the Neumann formula, the mutual inductanceof the couple is expressed by the summation of the mutual inductances among them.

M = ∑i=A,B,C,D

∑j=a,b,c,d

Mij (41)

Mij represents the mutual inductance between the line i either of the line A to D and the line jeither of the line a to d. The mutual inductances between the perpendicular lines such as MAb

are all zero and those between the parallel lines including MAa are expressed by using (39).Hence the mutual inductance of the square loops is derived as

M = M‖

[

ℓ1, ℓ2, x,√

(ℓ− + y)2 + z2

]

− M‖

[

ℓ1, ℓ2, x,√

(ℓ+ + y)2 + z2

]

+M‖

[

ℓ1, ℓ2, y,√

(ℓ− − x)2 + z2

]

− M‖

[

ℓ1, ℓ2, y,√

(ℓ+ − x)2 + z2

]

+M‖

[

ℓ1, ℓ2,−x,√

(ℓ− − y)2 + z2

]

− M‖

[

ℓ1, ℓ2,−x,√

(ℓ+ − y)2 + z2

]

+M‖

[

ℓ1, ℓ2,−y,√

(ℓ− + x)2 + z2

]

− M‖

[

ℓ1, ℓ2,−y,√

(ℓ+ + x)2 + z2

]

(42)

Note that the abbreviation ℓ± is defined as

ℓ± ≡ ℓ1 ± ℓ2

2(43)

5.3 Experimental measurements

The measured wireless power transmission efficiencies with the three types of impedanceratio control are shown in Fig. 12. The theoretical curves are derived by using the measuredcoupling coefficient, quality factors and the theoretical formulas (14 - 16). Note that thefrequency of the power source is 13.44 MHz, which is the average of the resonant frequencyof the transmitting and receiving resonator.The measurement results of the pickup loop slide position are shown in Fig. 13. Thetheoretical, which means optimum, curve of the impedance ratio is derived by using themeasured coupling coefficient, quality factors and the theoretical conditions of the impedance


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s1 s2

(a) Couple of circuits with uniformcurrent distributions

l1

l2

#

$

(b) Two parallel wires of finite line

l2

(x, y, z)

(0, 0, 0)

l1

A

B

C

D

a

b

cd

x

yz

(c) Couple of parallel square loops

Fig. 11. Geometries applied for the Neumann formula to derive the mutual inductances

0

20

40

60

80

100

0 0.5 1 1.5 2 2.5

Tra

nsm

issio

n e

ffic

iency,

[%]

Relative distance, z/l

Two-sided control (measured)One-sided control (measured)

No control (measured)Two-sided control (theory)One-sided control (theory)

No control (theory)

Fig. 12. Transmission efficiency with the three types of impedance matching control schemes.The relative distance is the ratio of a transmission distance z = 200 - 400 mm and a side lengthℓ = 198 mm of the transmitters and the receiver.

matching. The conditions are stated as (10) for the two-sided control and as (17) forthe one-sided control. The measured value of the impedance ratio is estimated by usingthe measured pickup loop slide position and the theoretical formula of the impedancetransformation (37) with the calculated properties of the resonator and the pickup loop.Each measured value well agreed with each theoretical value. Especially in the range ofz/ℓ = 1.3 - 2.0 for the two-sided control, the wireless power transmission efficiency wasas high as 30 - 70 % even though the coupling coefficient is lower than 0.02. Hence theeffectiveness of wireless power transmission with magnetic resonance was verified in themid-range. The optimum pickup loop slide positions became smaller when the transmissiondistances got larger. In the result, the corresponding impedance ratios became smaller as the


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0

2

4

6

8

10

1 1.2 1.4 1.6 1.8 2 2.20

10

20

30

40

50

Imp

ed

an

ce

ratio

,r S

=r D

Pic

ku

plo

op

po

sitio

n,

yS

=y

D[m

m]


Impedance ratio (measured)Impedance ratio (theory)

Pickup loop position (measured)

(a) Two-sided control

0

2

4

6

8

10

1 1.2 1.4 1.6 1.8 2 2.20

10

20

30

40

50

Imp

ed

an

ce

ratio

,r S

Pic

ku

plo

op

po

sitio

n,

yS

[mm

]


Impedance ratio (measured)Impedance ratio (theory)

Pickup loop position (measured)

(b) One-sided control

Fig. 13. The impedance ratio and the pickup loop slide position

theory predicts. This is because the magnitude of the magnetic field inside the square loopis larger near the border and smaller near the center. The controllable transmission range isabout z/ℓ = 1.3 - 2.0 with the two-sided control and z/ℓ = 1.5 - 2.0 with the one-sided control.The three efficiencies were equal at relative distance z/ℓ = 2.0. When the transmissiondistance became shorter, efficiency with the no control rapidly dropped. On the other hand,efficiency with the one-sided control went up a little and soon became almost constant.These characteristics suggest a wireless power transmission system with magnetic resonancewill have to carefully choose a proper impedance matching control scheme according to themoving range of mobile receivers in practical use. In narrow transmission distance range,no control would be enough. But in wide transmission distance range, one-sided controlis needed at least. If the application needs much higher efficiency at shorter distances, thetwo-sided control should be implemented.

6. Conclusions

This chapter studied the feasibility of wireless power transmission with magnetic resonanceto mobile objects.The theory of magnetic resonance was analyzed not with the coupled-mode theory but withthe electrical engineering theory to emphasize the essential elements of magnetic resonance:


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high quality factor resonator and impedance matching system. The theory also derives thethree schemes of impedance matching control. Two-sided control produces the maximumefficiency while it would require higher hardware and software costs. No control willonly need a simple and low-cost system in exchange for efficiency at various transmissiondistances. One-sided control has middle characteristics between the first two.A transmitter and receiver system prototype was developed to verify the theory and to discussthe realizable performance of a compactly-implemented resonator and impedance matchingsystem. The resonator was a loop with capacitor type resonator and the impedance matchingsystem was a sliding pickup loop system.Evaluation of the resonator quality factor showed the loop with capacitor type resonator hada higher quality factor than the other compactly-shaped dense coil type resonator. And itwas proven that the quality factor depends not only on radiation and ohmic loss but also oncapacitor loss.The theoretical analysis of the sliding pickup loop system and the power transmissionexperiment were explained. It showed that the couple of the pickup loop and the resonatorfunctions as an inductive transformer and the sliding position of the pickup loop controls theimpedance ratio. The power transmission experiment also verified the theory of wirelesspower transmission with magnetic resonance and the theoretical characteristics of treeimpedance matching control schemes.

7. References

A. Karalis, J. D. Joannopoulos & M. Soljacic (2008). Efficient wireless non-radiative mid-rangeenergy transfer, Annals of Physics Volume 323, Issue 1: 34–48.

A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher & M. Soljacic (2007). WirelessPower Transfer via Strongly Coupled Magnetic Resonances, Science Magazine Volume317(No. 5834): 83–86.

Carl T.A. Johnk (1975). Engineering Electromagnetic Fields and Waves, Wiley, New YorkD. M. Pozer (1998). Microwave Engineering, 2nd ed., Wiley, NYFrederick Warren Grover (1946). Inductance Calculations: Working Formulas and Tables, Van

Nostrand, New YorkH. Ayano, H. Nagase & H. Inaba (2003). High Efficient Contactless Electrical Energy

Transmission System, IEEJ Tran. IA Volume 123(No. 3): 263–270.T. Komaru, M. Koizumi, K. Komurasaki, T. Shibata & K. Kano (2010). Parametric Evaluation

of Mid-range Wireless Power Transmission with Magnetic Resonance, Proceedings ofthe IEEE-ICIT 2010 International Conference on Industrial Technology, pp. 789–792.

N. Ehara, Y. Nagatsuka, Y. Kaneko, S. Abe, T. Yasuda & K. Ida (2007). Compact andRectangular Transformer of Contactless Power Transfer System for Electric Vehicle,The Papers of Technical Meeting on Vehicle Technology, pp. 7–12.


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Wireless Power Transfer - Principles and Engineering ExplorationsEdited by Dr. Ki Young Kim

ISBN 978-953-307-874-8Hard cover, 272 pagesPublisher InTechPublished online 25, January, 2012Published in print edition January, 2012

InTech EuropeUniversity Campus STeP Ri Slavka Krautzeka 83/A 51000 Rijeka, Croatia Phone: +385 (51) 770 447 Fax: +385 (51) 686 166www.intechopen.com

InTech ChinaUnit 405, Office Block, Hotel Equatorial Shanghai No.65, Yan An Road (West), Shanghai, 200040, China

Phone: +86-21-62489820 Fax: +86-21-62489821

The title of this book, Wireless Power Transfer: Principles and Engineering Explorations, encompasses theoryand engineering technology, which are of interest for diverse classes of wireless power transfer. This book is acollection of contemporary research and developments in the area of wireless power transfer technology. Itconsists of 13 chapters that focus on interesting topics of wireless power links, and several system issues inwhich analytical methodologies, numerical simulation techniques, measurement techniques and methods, andapplicable examples are investigated.

How to referenceIn order to correctly reference this scholarly work, feel free to copy and paste the following:

Takashi Komaru, Masayoshi Koizumi, Kimiya Komurasaki, Takayuki Shibata and Kazuhiko Kano (2012).Compact and Tunable Transmitter and Receiver for Magnetic Resonance Power Transmission to MobileObjects, Wireless Power Transfer - Principles and Engineering Explorations, Dr. Ki Young Kim (Ed.), ISBN:978-953-307-874-8, InTech, Available from: http://www.intechopen.com/books/wireless-power-transfer-principles-and-engineering-explorations/compact-and-tunable-transmitter-and-receiver-for-magnetic-resonance-power-transmission-to-mobile-obj

© 2012 The Author(s). Licensee IntechOpen. This is an open access articledistributed under the terms of the Creative Commons Attribution 3.0License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

http://creativecommons.org/licenses/by/3.0

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